Assumptions REF  and REF  are minimal conditions for inference on precision matrices. Furthermore, they imply the conditions needed for clustering consistency derived in [1]} and discussed in Section REF , for \(n\)  large enough. The latter only require that \(\lambda _{\min }\left(_{  }^*\right)\)  is bounded from below by a sequence that converges to zero, as soon as \(\Vert ^*\Vert _{\max }\)  and \(\Vert ^*\Vert _{\max }\)  are bounded. This is strengthened by our assumptions. In particular, a constant lower bound on \(_{  }^*\)  is standard in any inference on graphical models and needed to show the asymptotic normality of the estimator introduced above [2]}, [3]}, [4]}.
